Questions — AQA (3508 questions)

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AQA AS Paper 2 2024 June Q4
3 marks Easy -1.3
4 Curve \(C\) has equation \(y = 8 \sin x\) 4
  1. Curve \(C\) is transformed onto curve \(C _ { 1 }\) by a translation of vector \(\left[ \begin{array} { l } 0 \\ 4 \end{array} \right]\)
    Find the equation of \(C _ { 1 }\) 4
  2. \(\quad\) Curve \(C\) is transformed onto curve \(C _ { 2 }\) by a stretch of scale factor 4 in the \(y\) direction. Find the equation of \(C _ { 2 }\) 4
  3. Curve \(C\) is transformed onto curve \(C _ { 3 }\) by a stretch of scale factor 2 in the \(x\) direction. Find the equation of \(C _ { 3 }\)
AQA AS Paper 2 2024 June Q5
3 marks Moderate -0.8
5 A student suggests that for any positive integer \(n\) the value of the expression $$4 n ^ { 2 } + 3$$ is always a prime number.
Prove that the student's statement is false by finding a counter example.
Fully justify your answer.
AQA AS Paper 2 2024 June Q6
7 marks Standard +0.3
6 In the expansion of \(( 3 + a x ) ^ { n }\), where \(a\) and \(n\) are integers, the coefficient of \(x ^ { 2 }\) is 4860 6
  1. Show that $$3 ^ { n } a ^ { 2 } n ( n - 1 ) = 87480$$ [3 marks]
    6
  2. The constant term in the expansion is 729 The coefficient of \(x\) in the expansion is negative. 6
    1. Verify that \(n = 6\)
      6
  3. (ii) Find the value of \(a\)
AQA AS Paper 2 2024 June Q7
9 marks Moderate -0.3
7
  1. Find the equation of the perpendicular bisector of \(A B\)
    7
  2. \(\quad\) A circle passes through the points \(A\) and \(B\) A diameter of the circle lies along the \(x\)-axis.
    Find the equation of the circle.
    [0pt] [4 marks]
AQA AS Paper 2 2024 June Q8
5 marks Standard +0.3
8 Prove that the graph of the curve with equation $$y = x ^ { 3 } + 15 x - \frac { 18 } { x }$$ has no stationary points.
[0pt] [5 marks]
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-11_2491_1753_173_123}
AQA AS Paper 2 2024 June Q9
9 marks Moderate -0.3
9 A curve has equation $$y = x - a \sqrt { x } + b$$ where \(a\) and \(b\) are constants. The curve intersects the line \(y = 2\) at points with coordinates \(( 1,2 )\) and \(( 9,2 )\), as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-12_696_807_641_605} 9
  1. Show that \(a\) has the value 4 and find the value of \(b\)
    9
  2. On the diagram, the region enclosed between the curve and the line \(y = 2\) is shaded. Show that the area of this shaded region is \(\frac { 16 } { 3 }\) Fully justify your answer.
    [0pt] [6 marks]
AQA AS Paper 2 2024 June Q10
11 marks Moderate -0.3
10
  1. (ii) Using the graph, estimate the value of the constant \(a\) and the value of the constant \(k\) [4 marks]
    \hline \end{tabular} \end{center} 10
    1. Show that \(\frac { \mathrm { d } F } { \mathrm {~d} t } = k F\)
      10
  2. (ii) Using the model, estimate the rate at which the number of followers is increasing 5 days after the song is released.
    10
  3. The singer claims that 30 days after the song is released, the account will have more than a billion followers. Comment on the singer's claim.
AQA AS Paper 2 2024 June Q11
1 marks Easy -1.8
11 The table below shows the daily salt intake, \(x\) grams, and the daily Vitamin C intake, \(y\) milligrams, for a group of 10 adults.
AdultABCDEFGHIJ
\(\boldsymbol { x }\)5.36.23.610.42.49.4657.111.2
\(y\)9014588481144480955541
A scatter diagram of the data is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-17_675_1150_1110_431} One of the adults is an outlier. Identify the letter of the adult that is the outlier.
Circle your answer below.
A
B
E
J Which one of the following is not a measure of spread?
Circle your answer.
median
range
standard deviation
variance
AQA AS Paper 2 2024 June Q13
4 marks Easy -1.8
13 The headteacher of a school wishes to collect the opinions of the students on a new timetable structure. To do this, a random sample of size 50 , stratified by year group, will be selected.
The school has a total of 720 students. The number of students in each of the year groups at this school is shown below.
Year group10111213
Number of students200240150130
13
  1. Find the number of students from each year group that should be selected in the stratified random sample.
    13
  2. State one advantage of using a stratified random sample.
AQA AS Paper 2 2024 June Q14
4 marks Moderate -0.8
14 The discrete random variables \(X\) and \(Y\) can be modelled by the distributions $$\begin{gathered} X \sim \mathrm {~B} ( 40 , p ) \\ Y \sim \mathrm {~B} ( 25,0.6 ) \end{gathered}$$ It is given that the mean of \(X\) is equal to the variance of \(Y\) 14
  1. Find the value of \(p\)
    14
  2. \(\quad\) Find \(\mathrm { P } ( Y = 17 )\)
AQA AS Paper 2 2024 June Q15
7 marks Easy -1.2
15 The number of flowers which grow on a certain type of plant can be modelled by the discrete random variable \(X\) The probability distribution of \(X\) is given in the table below.
\(\boldsymbol { x }\)012345 or more
\(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)0.030.150.220.310.09\(p\)
15
  1. Find the value of \(p\)
    15
  2. Two plants of this type are randomly selected from a large batch received from a local garden centre. Find the probability that the two plants will produce a total of three flowers.
    [0pt] [3 marks]
    15
    1. State one assumption necessary for the calculation in part (b) to be valid. 15
  3. (ii) Comment on whether, in reality, this assumption is likely to be valid.
AQA AS Paper 2 2024 June Q16
5 marks Easy -1.8
16
  1. (ii) &
    16
    where \(n\) is the total number of cars which had a measured hydrocarbon emission in the Large Data Set.
    16
  2. Find the mean of \(X\)
    [1 mark]
    16

  3. \hline &
    \hline \end{tabular} \end{center} 16
  • (ii) State one type of emission where more than 80\% of the data is known for cars in the entire UK Department for Transport Stock Vehicle Database.
    [0pt] [1 mark]
  • AQA AS Paper 2 2024 June Q17
    5 marks Moderate -0.3
    17 The proportion of vegans in a city is thought to be 8\% The owner of an organic food café in this city believes that the proportion of their customers who are vegan is greater than \(8 \%\) To test this belief, a random sample of 50 customers at the café were interviewed and it was found that 7 of them were vegan. Investigate, at the \(5 \%\) level, whether this sample supports the owner's belief.
    \includegraphics[max width=\textwidth, alt={}, center]{f5e0d980-4c50-4735-aea7-1bdf448a58f7-25_2498_1915_166_123}
    AQA AS Paper 2 Specimen Q1
    1 marks Easy -1.2
    1 \(\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 3 x + a\), where \(a\) is a constant.
    Given that \(x - 3\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\)
    Circle your answer.
    [0pt] [1 mark]
    \(- 9 - 339\)
    AQA AS Paper 2 Specimen Q2
    1 marks Easy -1.3
    2 The graph of \(y = \mathrm { f } ( x )\) is shown in Figure 1. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{f2bf5e19-98ba-4047-9023-3cfe20987e01-03_536_849_664_735}
    \end{figure} State the equation of the graph shown in Figure 2. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{f2bf5e19-98ba-4047-9023-3cfe20987e01-03_532_851_1530_733}
    \end{figure} Circle your answer.
    [0pt] [1 mark] $$y = \mathrm { f } ( 2 x ) \quad y = \mathrm { f } \left( \frac { x } { 2 } \right) \quad y = 2 \mathrm { f } ( x ) \quad y = \frac { 1 } { 2 } \mathrm { f } ( x )$$
    AQA AS Paper 2 Specimen Q3
    2 marks Easy -1.8
    3 Find the value of \(\log _ { a } \left( a ^ { 3 } \right) + \log _ { a } \left( \frac { 1 } { a } \right)\)
    [0pt] [2 marks]
    AQA AS Paper 2 Specimen Q4
    3 marks Easy -1.2
    4 Find the coordinates, in terms of \(a\), of the minimum point on the curve \(y = x ^ { 2 } - 5 x + a\), where \(a\) is a constant. Fully justify your answer.
    [0pt] [3 marks]
    AQA AS Paper 2 Specimen Q5
    4 marks Moderate -0.8
    5 The quadratic equation \(3 x ^ { 2 } + 4 x + ( 2 k - 1 ) = 0\) has real and distinct roots.
    Find the possible values of the constant \(k\)
    Fully justify your answer.
    [0pt] [4 marks]
    AQA AS Paper 2 Specimen Q6
    4 marks Moderate -0.8
    6 A curve has equation \(y = 6 x ^ { 2 } + \frac { 8 } { x ^ { 2 } }\) and is sketched below for \(x > 0\)
    \includegraphics[max width=\textwidth, alt={}, center]{f2bf5e19-98ba-4047-9023-3cfe20987e01-06_638_842_539_758} Find the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = a\) and \(x = 2 a\), where \(a > 0\), giving your answer in terms of \(a\)
    [0pt] [4 marks]
    AQA AS Paper 2 Specimen Q7
    5 marks Standard +0.3
    7 Solve the equation $$\sin \theta \tan \theta + 2 \sin \theta = 3 \cos \theta \quad \text { where } \cos \theta \neq 0$$ Give all values of \(\theta\) to the nearest degree in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\)
    Fully justify your answer.
    [0pt] [5 marks]
    AQA AS Paper 2 Specimen Q8
    6 marks Moderate -0.5
    8 Prove that the function \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 15 x - 1\) is an increasing function.
    [0pt] [6 marks]
    AQA AS Paper 2 Specimen Q9
    5 marks Moderate -0.8
    9 A curve has equation \(y = \mathrm { e } ^ { 2 x }\)
    Find the coordinates of the point on the curve where the gradient of the curve is \(\frac { 1 } { 2 }\) Give your answer in an exact form.
    [0pt] [5 marks]
    David has been investigating the population of rabbits on an island during a three-year period. Based on data that he has collected, David decides to model the population of rabbits, \(R\), by the formula $$R = 50 \mathrm { e } ^ { 0.5 t }$$ where \(t\) is the time in years after 1 January 2016.
    AQA AS Paper 2 Specimen Q10
    9 marks Moderate -0.8
    10
    1. Using David's model: 10
      1. state the population of rabbits on the island on 1 January 2016; 10
    2. (ii) predict the population of rabbits on 1 January 2021. 10
    3. Use David's model to find the value of \(t\) when \(R = 150\), giving your answer to three significant figures.
      [0pt] [2 marks] 10
    4. Give one reason why David's model may not be appropriate.
      [0pt] [1 mark] 10
    5. On the same island, the population of crickets, \(C\), can be modelled by the formula $$C = 1000 \mathrm { e } ^ { 0.1 t }$$ where \(t\) is the time in years after 1 January 2016.
      Using the two models, find the year during which the population of rabbits first exceeds the population of crickets.
      [0pt] [3 marks]
    AQA AS Paper 2 Specimen Q11
    10 marks Moderate -0.3
    11 The circle with equation \(( x - 7 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 5\) has centre \(C\). 11
      1. Write down the radius of the circle. 11
    1. (ii) Write down the coordinates of \(C\).
      [0pt] [1 mark] 11
    2. The point \(P ( 5 , - 1 )\) lies on the circle.
      Find the equation of the tangent to the circle at \(P\), giving your answer in the form \(y = m x + c\)
      [0pt] [4 marks] 11
    3. The point \(Q ( 3,3 )\) lies outside the circle and the point \(T\) lies on the circle such that \(Q T\) is a tangent to the circle. Find the length of \(Q T\).
      [0pt] [4 marks]
    AQA AS Paper 2 Specimen Q12
    4 marks Moderate -0.3
    12
    1. Given that \(n\) is an even number, prove that \(9 n ^ { 2 } + 6 n\) has a factor of 12
      [0pt] [3 marks]
      12
    2. Determine if \(9 n ^ { 2 } + 6 n\) has a factor of 12 for any integer \(n\).
      END OF SECTION A